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Affine processes under parameter uncertainty

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  • Tolulope Fadina
  • Ariel Neufeld
  • Thorsten Schmidt

Abstract

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non-linear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in the parameter set. This non-linear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity. We then develop an appropriate Ito-formula, the respective term-structure equations and study the non-linear versions of the Vasicek and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear Vasicek-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence the approach solves this modelling issue arising with negative interest rates.

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  • Tolulope Fadina & Ariel Neufeld & Thorsten Schmidt, 2018. "Affine processes under parameter uncertainty," Papers 1806.02912, arXiv.org, revised Mar 2019.
    • Handle: RePEc:arx:papers:1806.02912
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    References listed on IDEAS

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    1. Larry G. Epstein & Shaolin Ji, 2013. "Ambiguous Volatility and Asset Pricing in Continuous Time," The Review of Financial Studies, Society for Financial Studies, vol. 26(7), pages 1740-1786.
    2. Rama Cont, 2006. "Model Uncertainty And Its Impact On The Pricing Of Derivative Instruments," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 519-547, July.
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    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    5. Barrieu, Pauline & Scandolo, Giacomo, 2015. "Assessing financial model risk," European Journal of Operational Research, Elsevier, vol. 242(2), pages 546-556.
    6. B. Acciaio & M. Beiglböck & F. Penkner & W. Schachermayer, 2016. "A Model-Free Version Of The Fundamental Theorem Of Asset Pricing And The Super-Replication Theorem," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 233-251, April.
    7. Paul Wilmott & Asli Oztukel, 1998. "Uncertain Parameters, an Empirical Stochastic Volatility Model and Confidence Limits," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 175-189.
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    Citations

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    Cited by:

    1. Julian Hölzermann, 2024. "Pricing interest rate derivatives under volatility uncertainty," Annals of Operations Research, Springer, vol. 336(1), pages 153-182, May.
    2. Sandrine Gumbel & Thorsten Schmidt, 2020. "Machine learning for multiple yield curve markets: fast calibration in the Gaussian affine framework," Papers 2004.07736, arXiv.org, revised Apr 2020.
    3. Benedikt Geuchen & Katharina Oberpriller & Thorsten Schmidt, 2022. "Affine models with path-dependence under parameter uncertainty and their application in finance," Papers 2207.13350, arXiv.org, revised Jun 2024.
    4. Schmidt, Thorsten & Tappe, Stefan & Yu, Weijun, 2020. "Infinite dimensional affine processes," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7131-7169.
    5. Meriam El Mansour & Emmanuel Lepinette, 2023. "Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty," Papers 2311.08847, arXiv.org.
    6. Biagini, Francesca & Mazzon, Andrea & Oberpriller, Katharina, 2023. "Reduced-form framework for multiple ordered default times under model uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 1-43.
    7. Tolulope Fadina & Thorsten Schmidt, 2024. "The Unfairness of $\varepsilon$-Fairness," Papers 2405.09360, arXiv.org, revised Jun 2024.
    8. Eva Lutkebohmert & Thorsten Schmidt & Julian Sester, 2021. "Robust deep hedging," Papers 2106.10024, arXiv.org, revised Nov 2021.
    9. Criens, David & Niemann, Lars, 2024. "A class of multidimensional nonlinear diffusions with the Feller property," Statistics & Probability Letters, Elsevier, vol. 208(C).
    10. Tolulope Fadina & Thorsten Schmidt, 2019. "Default Ambiguity," Risks, MDPI, vol. 7(2), pages 1-17, June.
    11. David Criens & Lars Niemann, 2022. "Robust utility maximization with nonlinear continuous semimartingales," Papers 2206.14015, arXiv.org, revised Aug 2023.
    12. David Criens & Lars Niemann, 2023. "Robust utility maximization with nonlinear continuous semimartingales," Mathematics and Financial Economics, Springer, volume 17, number 5, September.
    13. Fuhrmann, Sven & Kupper, Michael & Nendel, Max, 2021. "Wasserstein Perturbations of Markovian Transition Semigroups," Center for Mathematical Economics Working Papers 649, Center for Mathematical Economics, Bielefeld University.
    14. Sandrine Gümbel & Thorsten Schmidt, 2020. "Machine Learning for Multiple Yield Curve Markets: Fast Calibration in the Gaussian Affine Framework," Risks, MDPI, vol. 8(2), pages 1-18, May.
    15. Hölzermann, Julian, 2020. "Pricing Interest Rate Derivatives under Volatility Uncertainty," Center for Mathematical Economics Working Papers 633, Center for Mathematical Economics, Bielefeld University.
    16. Blessing, Jonas & Kupper, Michael & Nendel, Max, 2023. "Convergence of Infintesimal Generators and Stability of Convex Montone Semigroups," Center for Mathematical Economics Working Papers 680, Center for Mathematical Economics, Bielefeld University.
    17. Akhtari, Bahar & Biagini, Francesca & Mazzon, Andrea & Oberpriller, Katharina, 2023. "Generalized Feynman–Kac formula under volatility uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    18. Changhong Guo & Shaomei Fang & Yong He, 2023. "A Generalized Stochastic Process: Fractional G-Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-34, March.
    19. Criens, David & Niemann, Lars, 2024. "Markov selections and Feller properties of nonlinear diffusions," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    20. Bahar Akhtari & Francesca Biagini & Andrea Mazzon & Katharina Oberpriller, 2020. "Generalized Feynman-Kac Formula under volatility uncertainty," Papers 2012.08163, arXiv.org, revised Nov 2022.
    21. Francesca Biagini & Georg Bollweg & Katharina Oberpriller, 2022. "Non-linear Affine Processes with Jumps," Papers 2207.03710, arXiv.org, revised Jul 2022.

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